A mathematical model for the deposition of particles from a thin sessile droplet undergoing diffusion-limited evaporation in four different modes of evaporation, namely the constant contact radius (CR), constant contact angle (CA), stick–slide (SS), and stick–jump (SJ) modes, is formulated and analysed. Explicit expressions are obtained for the flow and concentration of particles within the droplet, as well as the evolutions of the mass of particles in the bulk of the droplet and in a distributed deposit and/or in one or more ring deposits on the substrate. It is shown that the nature of the deposit depends on both the local evaporative flux and the motion of the contact line. In particular, for a droplet undergoing diffusion-limited evaporation, the flow is outwards towards the contact line in both the CR and CA modes, however, the receding contact line in the CA mode results in a qualitatively different deposit from that in the CR mode, specifically a switch from a ring deposit in the CR mode to a near-uniform deposit in the CA mode. This contrasts with the behaviour of a droplet undergoing spatially uniform evaporation in the CA mode, in which the flow is radially inwards resulting in a peak deposit. For a droplet evaporating in the SS or SJ modes, the final deposit is a combination of the deposit types associated with the CR and CA modes. The present model is validated by finding good agreement between the theoretical predictions for the deposit and previous experimental results.

We investigate droplet deformation following laser-pulse impact at low Weber numbers (${\textit{We}}\sim 0.1{-}100$). Droplet dynamics can be characterised by two key parameters: the impact Weber number and the width, W, of the distribution of the impact force over the droplet surface. By varying laser-pulse energy, our experiments traverse a phase space comprising (i) droplet oscillation, (ii) breakup or (iii) sheet formation. Numerical simulations complement the experiments by determining the pressure width and by allowing We and W to be varied independently, despite their correlation in the experiments. A single phase diagram, integrating observations from both experiments and simulations, demonstrates that all phenomena can be explained by a single parameter: the deformation Weber number ${\textit{We}}_{{d}}=f({\textit{We}},{W})$ that is based on the initial radial expansion speed of the droplet, following impact. The resulting phase diagram separates (i) droplet oscillation for ${\textit{We}}_{{d}}\lt 5$, from (ii) breakup for $5\lt {\textit{We}}_{{d}}\lt 60$ and (iii) sheet formation for ${\textit{We}}_{{d}}\gt 60$.

The chemical step is an elementary pattern in chemically heterogeneous substrates, featuring two regions of different wettability separated by a sharp border. Within the framework of lubrication theory, we investigate droplet motion and the contact-line dynamics driven by a chemical step, with the contact-line singularity addressed by the Navier slip condition. For both two-dimensional (2-D) and three-dimensional (3-D) droplets, two successive stages are identified: the migration stage, when the droplet traverses both regions, and the asymmetric spreading stage, when the droplet spreads on the hydrophilic region while being constrained by the border. For 2-D droplets, we present a matched asymptotic analysis that agrees with numerical solutions. In the migration stage, a 2-D droplet can exhibit translational motion with a constant speed. In the asymmetric spreading stage, the contact line at the droplet rear is pinned at the border. We show that a boundary layer still exists near the pinned contact line, across which the slope is approximately constant, whereas the curvature would diverge in the absence of slip. For 3-D droplets, our numerical simulations show that the evolution is qualitatively analogous to the 2-D case, while being significantly affected by the lateral flow. The droplet length and width exhibit non-monotonic variations due to the lateral flow. Eventually, the droplet detaches from the border and approaches equilibrium at the hydrophilic substrate. Additionally, we demonstrate that the variation of the apparent contact angle at the instant of border contact only affects the early stage of droplet migration.

This study elucidates the influence of liquid viscosity on the hydrodynamics of simultaneous and non-simultaneous droplet-pair impacts on solid substrates. Using synchronised high-speed imaging and quantitative analysis, the spreading dynamics of droplet lamellae and their interaction-driven central sheet evolution are examined across a range of viscosities from 1.01 to 91.46 mPa s, representing Ohnesorge numbers of 0.002–0.177, under controlled impact Weber numbers in the range of 81–131 and dimensionless inter-droplet spacings in the range of 1.43–1.85. The findings reveal that increasing viscosity results in thicker lamella fronts, reduced spreading and a lower maximum central sheet height. In addition, the central sheet morphology transitions from ‘semilunar’ sheets to ephemeral liquid bumps, accompanied by suppressed capillary waves and reduced rim instabilities. A novel scaling law is derived for the maximum sheet extension, demonstrating its robust applicability to both simultaneous and non-simultaneous impacts of droplet pairs across varying viscosities and impact conditions. Furthermore, distinct morphological differences emerge between simultaneous and non-simultaneous impacts, primarily governed by lamella–lamella interactions and the momentum transfer dynamics. These findings enhance our understanding of the interplay between viscous and inertial forces in droplet-pair impacts, offering valuable insights for optimising spray-based technologies and multiphase fluid systems.

While surfactants are known to affect fluid–fluid interfaces, their impact on solid–liquid interfaces is an open problem. Here, we show that surfactants carried by a spreading nonpolar droplet can dynamically alter the solid–liquid interfacial energy, leading to a new spreading regime beyond the classical Tanner’s law and known Marangoni regimes. We develop a theoretical framework that combines the new spreading mechanism, governed by the solid–liquid interfacial energy gradient, together with capillarity. Experiments across twelve distinct combinations of nonpolar solvents, surfactants, and substrates confirm our theoretical predictions for the transition from Tanner’s law to the newly uncovered spreading regime. Our findings provide predictive control for applications in coatings, printing, microfluidics and surface engineering.

The evaporation of multicomponent sessile droplets is key in many physicochemical applications such as inkjet printing, spray cooling and micro-fabrication. Past fundamental research has primarily concentrated on single drops, though in applications they are rarely isolated. Here, we experimentally explore the effect of neighbouring drops on the evaporation process, employing direct imaging, confocal microscopy and particle tracking velocimetry. Remarkably, the centres of the drops move away from each other rather than towards each other, as we would expect due to the shielding effect at the side of the neighbouring drop and the resulting reduced evaporation on that side. We hypothesise that pinning-induced motion mediated by suspended particles in the droplets (due to contamination or added on purpose) is the cause of this counter-intuitive behaviour. We also discuss an alternative interpretation, namely that the repulsion between the two droplets is caused by thermal Marangoni flow as is the case for a pair of pure droplets on an isothermal substrate (Malachtari and Karapetsas, J. Fluid Mech. vol. 978, 2024, p. A8), but give the arguments why that interpretation is not applicable in our case of binary droplets. To further support our interpretation, with the help of direct numerical simulations we explore the relative contributions of the replenishing flow and of the solutal and thermal Marangoni flows to the overall flow dynamics in one droplet. Finally, as further evidence, the azimuthal dependence of the radial velocity in the drop is compared with the evaporative flux and a perfect agreement is found.

In this paper, the evaporation of neighbouring multi-component droplets or rivulets – often found in applications such as inkjet printing, spray cooling and pesticide delivery – is studied numerically and theoretically. The proximity induces a shielding effect that reduces individual evaporation rates and disrupts the symmetry of both the concentration profile and the flow field in the liquids. We examine how the symmetry of flow and concentration fields is affected by key parameters, namely the contact angle, the inter-droplet (or inter-rivulet) distance and the magnitude of surface tension gradient forces (i.e. the Marangoni number). We focus on binary mixtures, such as water and 1,2-hexanediol, where only one component evaporates and evaporation is slow, thereby allowing simplifications to the governing equations. To manage the complexity of the full three-dimensional droplet problem, we begin with a two-dimensional model of neighbouring rivulets. Solving the complete transient equations for rivulets with pinned contact lines and fixed inter-rivulet distance reveals that the asymmetry – quantified by the position of the interfacial stagnation point of the flow – diminishes over time. Using a validated quasi-stationary model, we find, with increasing contact angle and inter-rivulet distance, that the stagnation point migrates closer to the centre, yet it remains unaffected by the Marangoni number. A simplified lubrication model applied to droplets shows similar dependencies on contact angle and distance, although here the stagnation point appears to vary with the Marangoni number. We attribute this dependence to the additional azimuthal flow in droplets, leading to a nonlinear evolution of the concentration and therefore a non-trivial dependence of the symmetry on the Marangoni number.

We perform axisymmetric numerical simulations to investigate the coalescence dynamics of a liquid drop in a deep liquid pool. This study aims to generalise the mechanisms of partial coalescence across a range of drop shapes, elucidate the underlying mechanism of neck oscillations, and examine the roles of inertial, viscous and gravitational forces, quantified by the Weber, Ohnesorge and Bond numbers, in governing the coalescence behaviour. A phase diagram is constructed to delineate the boundaries between partial and complete coalescence regimes based on these dimensionless parameters. Our analysis of the height-to-neck ratio shows that, upon contact with the pool, the primary drop forms an upward liquid column that ultimately pinches off due to inwardly directed horizontal momentum. Additionally, the study suggests that as the dimensionless numbers increase, the effect of the vertical collapse rate plays a significant role in the outcome of the coalescence process. Notably, the Rayleigh–Plateau instability is found to be insignificant in driving partial coalescence within the explored parameter space. We identified a transition regime between partial and complete coalescence, characterised by multiple neck oscillations that delay the pinch-off of secondary droplets. The formation of secondary droplets is most prominent for prolate drops, followed by spherical and oblate drops of comparable volume. Furthermore, we observe that the tendency to form multiple droplets from elongated liquid columns diminishes with an increase in the impact velocity of the primary drop.

Impact pressure temporal and spatial profiles caused by a droplet on a surface can assist in understanding of microscale erosion mechanisms. We derive analytical solutions of pressure profiles and impact force time-series on a surface for early and intermediate impact times, up to dimensionless time $0.27$, using unsteady potential flow. The solutions reproduce the ‘ring pattern’ of surface pressure at early impact and the temporal evolution to a shift to, at intermediate time, the centred maximum. This is due to the emerging dominance of the steady component of the Bernoulli equation. The analytical solutions use a model of an unsteady disk in infinite liquid to induce flows similar to those within impacting drops. We identify a separation point close to the expanding edge of the impacted droplet, which interprets the Wagner condition for droplet impact and extends the validity of the analytical wet radius from time $\textit {O}(10^{-2})$ in the literature to beyond $\textit {O}(1)$. This separation point resolves singularity issues at the wet radius and solutions address the droplet impact problem. The theoretical predictions agree with high-fidelity numerical calculations of the impact pressure at the surface centre, middle and at the radius of the ring pressure, and with the impact force for more than four time decades. The theoretical predictions remain at least qualitatively correct in dimensionless time for more than $\textit {O}(1)$. The analytical solutions are closed and explicit functions of space, time, wet radius expanding velocity and incidence angle, and provide ready estimation of droplet impact loadings for erosion problems.

We present a comprehensive experimental and theoretical investigation of the evaporation dynamics of freely levitated water droplets in an upward airstream under varying temperature and relative humidity conditions, using a custom-designed wind tunnel that replicates natural rainfall scenarios. A high-speed imaging system captures the temporal evolution of morphology, shape oscillations and size reduction of the droplet undergoing evaporation. Our observations reveal that larger droplets exhibit persistent shape oscillations due to the interplay between inertia and surface tension in the presence of convective airflow, which significantly alters the evaporation rate compared with that of a stationary spherical droplet in quiescent air. To quantify the effects of air convection, complex morphology and shape oscillations of the levitated droplet at different temperatures and humidity, we develop a modified evaporation model that extends the classical $d^2$ law. This model incorporates (i) a generalised Sherwood number that accounts for the variation in Reynolds number, Schmidt number, temperature and relative humidity; and (ii) a shape factor that captures the time-averaged surface area of oscillating droplets. The model is validated against experimental findings across a wide range of droplet sizes and environmental conditions, showing excellent agreement in predicting the temporal evolution of droplet diameter and total evaporation time. Furthermore, we construct a regime map showing the variation in the lifetime of the droplet in the temperature–humidity space. The present study establishes a framework that integrates convective transport and morphological deformation, offering new insights into the microphysics of raindrop evaporation.

Steady tip streaming in the limit of vanishing flow rate has been experimentally and numerically documented, yet theoretical solutions describing local conical Stokes flows have remained elusive. Here, we derive approximate analytical solutions for local conical flows in liquid–liquid flow focusing scenarios, addressing the limit of negligible emitted flow rate. Our analysis demonstrates the existence of a universal relationship between the inner-to-outer liquid viscosity ratio and the cone angle, establishing the theoretical underpinning for precise control of microscopic jet formation. A posteriori comparison with previously published experiments reveals that digitised cusp-like meniscus profiles collapse quantitatively onto the predicted slender-body similarity solution. These findings pave the way for technologies that require exact manipulation of fluid flows at nearly molecular dimensions.

We present a method to simulate non-coalescing impacts and rebounds of droplets onto the free surface of a liquid bath, together with new experimental data, focused on the low-speed impact of droplets. The method is derived from first principles and imposes only natural geometric and kinematic constraints on the motion of the impacting interfaces, yielding predictions for the evolution of the contact area, pressure distribution and wave field generated on both impacting masses. This work generalises an existing kinematic-match method whose prior applications dealt with deformation of the surface of the bath only; i.e. neglecting that of the droplet. The method’s extension to include droplet deformation gives predictions that compare favourably with existing experimental results and our new experiments conducted in the low-Weber-number regime.

Upon radial liquid sheet expansion, a bounding rim forms, with a thickness and stability governed, in part, by the liquid influx from the unsteady connected sheet. We examine how the thickness and fragmentation of such a radially expanding rim change upon its severance from its sheet, absent of liquid influx. To do so, we design an experiment enabling the study of rims pre- and post-severance by vaporising the thin neck connecting the rim. No vaporisation occurs of the bulk rim itself. We confirm that the severed rim follows a ballistic motion, with a radial velocity inherited from the sheet at severance time. We identify that the severed rim undergoes fragmentation in two types of junctions: the base of inherited, pre-severance, ligaments and the junction between nascent rim corrugations, with no significant distinction between the two associated time scales. The number of ligaments and fragments formed is captured well by the theoretical prediction of rim corrugation and ligament wavenumbers established for unsteady expanding sheets upon droplet impact on surfaces of comparable size to the droplet. Our findings are robust to changes in impacting laser energy and initial droplet size. Finally, we report and analyse the re-formation of the rim on the expanding sheet and propose a prediction for its characteristic corrugation time scale. Our findings highlight the fundamental mechanisms governing interfacial destabilisation of connected fluid-fed expanding rims that become severed, thereby clarifying destabilisation of freely radially expanding toroidal fluid structures absent of fluid influx.

Flexible substrates are effective in suppressing splashing, but they simultaneously lead to inhibition of spreading (Howland et al. 2016 Phys. Rev. Lett. vol. 117, 184502; Vasileiou et al. 2016 Proc. Natl Acad. Sci. USA vol. 113, pp. 13307−13312). In addition, there has been limited investigation and no established scaling law for the splashing threshold in the case of flexible substrates. To address these points, this paper proposes a lotus-leaf-like disk that can effectively suppress droplet splashing without inhibiting the maximum spreading of droplets. This situation is numerically studied in this paper. Five dynamic modes of the impacting droplet are identified with various Weber numbers (defined as the inertia force relative to the surface tension force) and different disk’s stiffnesses. The threshold Weber number of splashing is developed by considering the flexibility of substrates. Finally, the results demonstrate that the proposed method not only suppresses the splashing but also maintains the maximum spreading.

A droplet impacting a deep fluid bath is as common as rain over the ocean. If the impact is sufficiently gentle, the mediating air layer remains intact, and the droplet may rebound completely from the interface. In this work, we experimentally investigate the role of translational bath motion on the bouncing to coalescence transition. Over a range of parameters, we find that the relative bath motion systematically decreases the normal Weber number required to transition from bouncing to merging. Direct numerical simulations demonstrate that the depression created during impact combined with the translational motion of the bath enhances the air-layer drainage on the upstream side of the droplet, ultimately favouring coalescence. A simple geometric argument is presented that rationalises the collapse of the experimental threshold data, extending what is known for the case of axisymmetric normal impacts to the more general three-dimensional scenario of interest herein.

We present a three-dimensional numerical study of the splashing dynamics of non-spherical droplets impacting a quiescent liquid film, covering a wide range of aspect ratios ($A_r$) and Weber numbers ($ \textit{We}$). The simulations reveal distinct impact dynamics, such as spreading, splashing type-1, splashing type-2 and canopy formation, which are delineated in a regime map constructed in the $A_r$–$ \textit{We}$ parameter space. Our results demonstrate that droplet morphology during the impact significantly influences crown evolution and splash initiation, with oblate drops promoting finger growth and fragmentation due to enhanced rim deceleration, while prolate drops tend to form canopies. We observe that the hole instability, which becomes more prominent at higher Weber numbers, arises from lamella rupture in the thinnest region of the film, located just beneath the crown rim. A linear stability analysis, supplemented by the temporal evolution of the crown obtained from the numerical simulations, adequately predicts the number of fingers formed along the crown rim by accounting for both Rayleigh–Plateau (RP) and Rayleigh–Taylor (RT) instabilities. The theoretical analysis demonstrates the dominant role of the RP instability in determining the number and wavelength of early undulations, with the RT instability serving to amplify the growth rate of the disturbances. Our findings highlight the critical role of the droplet shape in splash dynamics, which is relevant to a range of applications involving droplet impact.

We developed a two-phase lattice Boltzmann model by coupling the entropic multiple-relaxation-time (EMRT or KBC) collision operator enabling low fluid viscosity, with a source term (Wang et al. 2022, Phys. Rev. E vol. 105, no 4) to independently adjust surface tension. The coupling is implemented via the exact difference method (EDM), which allows full consideration of external-force effects on the entropic stabiliser in KBC, in contrast to the recent work of Wang et al. (2022 Phys. Rev. E vol. 105) and Xu et al. (2024 Comput. Math. Appl. vol. 159, 92–101). More importantly, we address a major drawback of the EDM by explicitly demonstrating how its high-order error terms influence the pressure tensor and surface tension. Using the developed model, we investigated droplet impact and splashing on a thin liquid film at a remarkably high Weber number of ${\textit{We}} = 5000$ and Reynolds number of ${\textit{Re}} = 5000$. Droplet impact and splashing on flat surfaces and mesh structures at very high ${\textit{Re}}$ (15 200) and ${\textit{We}}$ (1020) are also studied after validating four representative cases against experiments. For droplet impact on flat surfaces, hydrophobicity promotes the growth of peripheral instabilities, leading to fingering splashing. Corona splashing transitions to fingering splashing as the liquid–gas viscosity ratio increases. For droplet impact on mesh structures, large openings promote liquid penetration, whereas small openings enhance spreading. As the solid ratio increases, the maximum spreading ratio increases monotonically but nonlinearly, whereas the maximum penetrated liquid pillar length first rises and then drops. These simulations demonstrate the proposed model offers significant advantages for accurately capturing and elucidating complex droplet impact and splashing dynamics at high ${\textit{Re}}$ and ${\textit{We}}$.

A lattice Boltzmann method is adopted to investigate the breakup of surfactant-free and surfactant-laden droplets in both regular and irregular T-junction microchannels. During droplet neck contraction, the neck thinning shifts from inertia dominated to interfacial tension dominated, causing spontaneous rapid neck collapse due to Rayleigh–Plateau instability. For the regular rectangular microchannels, we find that the prerequisite for the spontaneous breakup of a surfactant-free droplet is that the local capillary pressure in the triggering area exceeds the Laplace pressure difference between the inside and outside of the droplet neck. Results show that the critical neck thickness $\delta _\textit{cr}^{*}$ for the droplet spontaneous breakup increases with increasing height-to-width ratio $\chi$ of the microchannel in both surfactant-free and surfactant-laden systems. The presence of surfactants decreases $\delta _\textit{cr}^{*}$ at the identified $\chi$, while the surfactant effects are gradually enhanced as $\chi$ increases. Subsequently, a constriction section is incorporated into the upper microchannel wall to establish an irregular microchannel. As constriction depth (length) increases, $\delta _\textit{cr}^{*}$ linearly decreases (increases) in the surfactant-free system, while $\delta _\textit{cr}^{*}$ exponentially decreases (linearly increases) in the surfactant-laden system. Four empirical formulas are proposed to predict the values of $\delta _\textit{cr}^{*}$ under varying constriction depths and lengths in the two systems.

In this work, we propose a lattice Boltzmann model (LBM) to simulate diverse particle deposition patterns induced by isothermal droplet evaporation. The model is composed of two distributions, for the multiphase flow with phase change, and the particle transport with deposition, coupled with a contact angle hysteresis model for the contact line stick-slip dynamics. The model is validated by two benchmarks, and our simulations agree well with the theoretical solutions or experimental results. With the validated LBM, we first reproduced diverse deposition patterns, ranging from the coffee ring, uniform, to mountain-type patterns in single and multiple symmetrical/unsymmetrical forms. Then a parametric study is conducted to investigate how the solvent/particle/substrate properties affect the evaporation dynamics and resultant deposition patterns. Afterwards, we apply the average ratio ($r_{\phi ,a}$) of particles deposited at the droplet periphery and the centre to quantitatively classify the diverse emerging patterns. We show that $r_{\phi ,a}$ is controlled by the competition between the capillary transport and particle diffusion, leading to a linear dependence on the average Péclet number $\textit{Pe}_{a}$. Finally, we validated the scaling by lattice Boltzmann simulations with the proposed $\textit{Pe}_{a}$ spanning over three orders of magnitude, supplemented by discussions from the aspect of the particle transport equation.

We consider the problem of a cylindrical (quasi-two-dimensional) droplet impacting on a hard surface. Cylindrical droplet impact can be engineered in the laboratory, and a theoretical model of the system can also be used to shed light on various complex experiments involving the impact of liquid sheets. We formulate a rim-lamella model for the droplet-impact problem. Using Gronwall’s inequality applied to the model, we establish theoretical bounds for the maximum spreading radius $\mathcal{R}_{\textit{max}}$ in droplet impact, specifically $k_1 {\textit{Re}}^{1/3}-k_2(1-\cos \vartheta _a)^{1/2}({\textit{Re}}/{\textit{We}})^{1/2}\leq \mathcal{R}_{\textit{max}}/R_0\leq k_1{\textit{Re}}^{1/3}$, valid for ${\textit{Re}}$ and ${\textit{We}}$ sufficiently large. Here, ${\textit{Re}}$ and ${\textit{We}}$ are the Reynolds and Weber number based on the droplet’s pre-impact velocity and radius $R_0$, $\vartheta _a$ is the advancing contact angle (assumed constant in our simplified analysis) and $k_1$ and $k_2$ are constants. We perform several campaigns of simulations using the volume of fluid method to model the droplet impact, and we find that the simulation results fall within the theoretical bounds.